Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{5y^2 - 405}{-2y^3 + 32y^2 - 126y}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {5(y^2 - 81)} {-2y(y^2 - 16y + 63)} $ $ p = -\dfrac{5}{2y} \cdot \dfrac{y^2 - 81}{y^2 - 16y + 63} $ Next factor the numerator and denominator. $ p = - \dfrac{5}{2y} \cdot \dfrac{(y - 9)(y + 9)}{(y - 9)(y - 7)}$ Assuming $y \neq 9$ , we can cancel the $y - 9$ $ p = - \dfrac{5}{2y} \cdot \dfrac{y + 9}{y - 7}$ Therefore: $ p = \dfrac{ -5(y + 9)}{ 2y(y - 7)}$, $y \neq 9$